LEVELFINITE ELEMENT ANALYSIS OF DYNAMIC CRACK PROPAGATION

I

by

J. Jung, J'. Ahmad and M. F. Kanninen

BattelleColumbus Laboratories

N.• Columbus, Ohio

and

L4LJ C. H. PopelarEngineering Mechanics Department

The Ohio State UniversityColumbus, Ohio

March 1981

To be presented at the Failure Prevention and Reliability Conference inHartford, Connecticut, September 20-23, 1981.

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FINITE ELEMENT ANALYSIS OF DYNAMIC CRACK InterimPROPAGATION *. PERFORMING ORG, REPORT NUMIER

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J. Jung, J. Ahmad, M. F. Kanninen and N00014-77-C-0576 - Battelle

C. H. Popelar

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Battelle's Columbus Laboratories505 King AvenueColumbus. Ohio 43201

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Fracture, Dynamic Crack Propagation, Finite Element Analysis

20. ABST!LT (Continue an fewere* eide if nececomy and identitr by block number)

A finite element analysis of stationary and propagating cracks in thepresence of inertia forces is presented. An extension of the J-integralapproach is employed. To model a propagating crack, a conceptually simpleyet effective technique has been developed.. The new crack propagation scheme

eliminates the difficulties associated with the use of moving singularelements.

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II

FINITE ELEMENT ANALYSIS OF DYNAMIC CRACK PROPAGATION

by

AccesIion-' For J. Jung, J. Ahmad and M. F. Kanninen

NTIS G1AA&IDTIC TAB BattelleUnannounced Columbus LaboratoriesJustification . . Columbus, Ohio

and

-Distributin/ .. ... C. H. Popelar

•Availability Codes Engineering Mechanics DepartmentAvai! and/or The Ohio State University

Dist Special Columbus, Ohio

ABSTRACT

A finite element analysis of stationary and propagating :racks in

the presence of inertia forces is presented. An extension of the J-integral

approach is employed. To model a propagating crack, a conceptually simple yet

effective technique has been developed. The new crack propagation scheme

eliminates the difficulties associated with the use of moving singular

elements.

INTRODUCTION

It is generally accepted that a crack arrest methodology based on a

dynamic view of crack propagation and arrest is more fundamental than the

* quasi-static approach to the problem [1). Specifictlly, when inertia forces,

stress-wave reflections, and rate-dependent fractur, processes are dominant,

quasi-static assumptions will generally underestimi te the true crack driving

force. In many applications, such as nuclear presture vessel design and

structures subjected to impact loading, dynamic ef'ects can be important. In

these cases analytical models based on the quasi-static assumption may lead to

erroneous conclusions.

2 'Inclusion of dynamic features in an analytical model undoubtedly

leads to complications. In recent years the finite element method has emerged

as an important tool that can be used to resolve at least some of the mathe-

matical difficulties associated with the problem. Considerable progress has

also been made toward understanding some of the basic concepts.

The purpose of the paper is to present the salient features of arecently developed finite element dynamic crack propagation modeling tech-nique. Results of an elasto-dynamic analyses will be presiented for both

stationary and running cracks. Comparisons will be made with available

experimental results. Through the analyses of test specimens, it will be

Aiemonstrated how this analytical model can be used to acquire a better

understanding of the dynamic crack propagation phenomenon.

Dynamic Analysis and Numerical Integration

The familiar finite element discretized version of the equations of

motion are simply written in the following form:

[M] {X} + [C] {X} + (K] {X} -{R}

where

[M] - mass matrix

[C] -damping matrix

[K] - stiffness matrix

{R} the external load vector

{X}, {X}, {X} the displacement, velocity, and acceleration vectors

respectively.

There are several methods that could be used to perform the direct numerical

integrations of the equations. The method selected for this work was the

Newmark implicit scheme [2]. For this method, the following approximations

are used:i,'

t+At t + 1(1-6) " t + Sxt + At At (2)

x+ xt + At + [(/2 -a) Xt +a )X + t (3)

where a and 6 are parameters that can be varied for accuracy and stability

while At is the time step. From the above equations it is possible to write

the equations of equilibrium for time, t + At, in terms of displacements, ve-

locities, and accelerations at time, t.

[[K] + aoIM] + .[C] Ixt + At - Rt + At + EMI (aotXt} + a + a

+ [CI(a1{Xt} + a 4 t} + a -'t}) (4)

where

a=--;a =- -- -;a ----0 2 - aAt -2 aAt -3 2ao

a -- 1 a5 i. ( 2) ;a 6 At (1-6) (5)

a7 -= At.

Solving Equation C4) yields IX t + At whereupon the accelerations and veloci-

ties at time, t + At, can be calculated using Equations (2) and (3).

Elasto-dynamic Analysis of Stationary Cracks

Consider the problem of determining the stress intensity factor for

a stationary crack in a structure subjected to dynamic loadings. Several in-

vestigators have solved this problem by employing singular elements around the

crack tip (3-5]. While this approach has proved to be successful, it will

later require special considerations when the crack is prcpagating. An alter-

native to the singular element approach is to derive the stress intensity fac-

tors from a path independent J-Integral [6]. After accounting for the inertia

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effects, the rate of energy release per unit of crack advance in the direction

of the crack X, is defined as:

- f arsa [wdx., -aijnj u dUi dA (6)Fr+r ax ds +f f A U ix)

where the first integral is the conventional J-integral over an arbitrary path

surrounding the crack tip and the second intigral is an integral over the

area, A, enclosed by the path, r + r.; see Figure 1.

An application of this J-integral is shown in the following example

in which a centrally cracked plate in plane strain is impulsively loaded tr a

uniform stress; see Figures 2 and 3. The finite element model employed 309

nodes and 90 eight noded quadratic isoparametric elements; see Figure 4. The

material is linear elastic with a Young's modulus of 200 GPa, Poisson's ratio

of 0.3 and density of 5 g/cm3 . Newmark's implicit time integration scheme was

used (a - 0.25 and 6 - 0.5) with a time step of 0.15 microseconds. A consis-L-

tent mass formulation was used for this analysis.

The results of the present analysis are shown in Figure 5 along with

those of other investigators for comparison [7,8]. As the figure shows, the

present analysis is in excellent agreement with the other results. It should

be mentioned that the path independencE of this J-integral has been previously

demonstrated [7].

An Analysis of Running Cracks

A technique has been developed that allows for the analysis of run-

ning cracks without the need for mesh adjustments or iterations. This tech-

nique is based on earlier crack propagation investigations using a finite dif-

ference based analysis [91. It begins by subdividing the element immediately

ahead of the crack tip into what can be thought of as subelements, as shown in

Figure 6, (in this case, for a mesh composed of four noded linear elements).

During propagation, the crack tip will be, in theory, allowed to move indiscrete jumps along the crack plane through these subelements; e.g., the

crack tip will move from Point "1" to Point "2" in one jump and later move

A: 5

FIGURE 1. COORDINATE SYSTEM AND CURVES r and r.S

crlt

iii

FIGURE 2. GEOMETRY FOR STATIONARY CRACK ANALYSIS;a=0.24, em.

' I

! I

6 I

alt

I

0 1

FG 3.LAIGHS ORYFO SAI ONRYCAC AAYSS

I

I

r - -

I1.II

SI

I - -. -I

C----- J- II tregl Poth

FIGURE 4. FINITE ELEMENT MESH AND i-INTEGRAL PATH FORA STATIONARY CRACK PROBLEM

I-.- -. ,~

S÷8

30

PR~s~,rr AMLYsIs

+1++4

Tim+, +irosecond

FIGURE 5. DYNAM(IC STRESS INTENSITY F'ACTOR VERSUS TIME FORAN IMPULSIVELY LOADED CENTER-CRACKED PANEL WITHA STATIONARY CRACK

S. . ._ -. • ,

9

ol .0

[3

FIGURE 6. ELEMENT SUBDIVISION.

r

10

from Point "2' to Point "3" and so on, as the analysis dictates. The stress

intensity factor at any crack location will be determined by a i-integral

evaluation, Equation (6).

Consider the situation with the crack tip at Point "1". The crack

velocity, V, will be estimated by counting the number of time steps, n, which

are required until the stress intensity factor, K, is equal to the fracture

toughness and performing the simple calculation:

V a x(2) - x(l)I (7)

where At is the time step size. This algorithm is repeated as the crack tipmoves from Point "1" to "2", "2" to "Y", and so on.

This algorithm has the advantage of allowing the calculation of

pseudo crack velocities after each time step. This pseudo velocity is equal

to the actual crack velocity when the fracture criterion

K -K (V, T)

D

* I is satisfied, where the fracture toughness can be a function of the crack

velocity, V, and material temperature, T. Since the crack velocity is calcu-

lated first and then the fracture toughness, there are no problems with using

a velocity independent fracture toughness relation. This type of criterion

would cause problems if the crack was propagating, i.e., K - KD, and the

inverse of the KD relation was used to determine the crack velocity as is done

in some codes.V It was then necessary to decide the number of subdivisions to make

in each element. This was accomplished by determining the maximum crackvelocity to be allowed in the analysis, ma* In this study, Vmxwas taken

14x Va

as the bar wave speed, VE p. Once Vmax is set, the maximum distance the crack

can propagate per time step, Ax', is

Ax, V eAt .(8)

max

AA

The number of Ax' units in an element length, Ax, is, Tx'" To make the number

of subdivisions an integer value and to prevent the crack from traveling more

d than one element length per time step, the number of subdivisions, N, is taken

to be

where N is a truncated integer value. Should the crack want to propagate at

•. one subdivision per time step, its propagation speed will be somewhat less

• ~than the maximum velocity, Vmax, set previously due to the truncation. The

i ~ amount by which the actual maximum crack velocity differs from Vmax Til; "

-4'.

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depend on the value of N.TThe last detail was to determine how the crack tip would be placed n

at the subdivision lines. This was conceptually performed by placing a force

on the element to which the crack tip is adjacent. For a four-noded element,

the force is placed on the one node on the crack plane behind the crack tipand for an eight-noded element nodal, forces would be placed on the two nodes

•- ' as shown in Figure 7. The forces were postulated to be linearly related tothe craclt tip location by the followi osg equation:

:;" 0i

where Fi is the force at node "i", Fo is the nodal force at node "i" just

• prior to the node release, as shown in Figure 7, "a" is the crack length inFthe element which the crack tip is in, and "Ax" is the length of that

element. In the present study, eight-noded quadratic isoparametric elements

were used. The midside node was released simultaneously with the trailingavertex node. The force history of each node followed the prescription given

in Equation (e0). Both nodes were released simultaneously to avoid possible

problems d~scovered in another investigation [101.

A summary of the above algorithm for a quasi-statically initiatedevent is given in Figure 8.

In the algorithm just described, the location of the "crack tep" is

io

ubviously not actually known when there are forces on the crack face. The

r

th lmn hc h raktpi n n A' stelnt fta

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(a).- a .- *-

C'Crac.k Tip

(b)

'-Crock Tip

FIGURE 7. PLACEMENT OF NODAL FORCES FOR (a) A FOUR-NODEDELEMENT AND (b) AN EIGHT-NODED ELEMENT.

13

"DEERIN INTA TTC OFGRTO

DETERMINE THE NUMBER OF SUBDIVISIONSIN THE CURRENT CRACK TIP ELEMENT

CALCULATE THE STRESS INTENSITY FACTOR, K1 "

CALCULATE THE CRACK VELOCITY, V

NO - 'K- t-t+Lt i

YES 1-

t-t+At [II -

PROPAGAT~E 3TECACK ONE ELEMENT SUBDIVISION1

TEST IF THE CRACK HAS REACHED NOTHE ELEMENT BOUNDARY

YES

FIGURE 8. FLOW CHART OF PROPAGATION ALGORITHM FOR AQUASI-STATICALLY INITIATED DYNAMIC EVENT

14

only time when the location of the crack tip is unambiguous is when there are

no forces on the crack face. But in order to avoid regeneration of the mesh

(which would be necessary to place a node at the desired crack tip location)

the interpretation used In the above algorithm was adopted. A similar inter-

pretation has also been used by others [10,11,12].

The applicability of this interpretation can be some-what tested by

performing a constant velocity crack analysis. The problem chosen was a cen-

trally cracked square plate of 40 mm x 40 mm with an initial crack length of

0.2 a/w, where w is the panel width. The panel was initially loaded with a

uniform stress in the direction perpendicular to the crack. The properties of

the plate were E - 7716 kgf/mm2 ; v = 0.286; and p - 2.5 x 10-10 kgf - s 2 /mm4 .

The crack was propagated at a velocity of 0.2 of the shear wave speed of the

material which was Cs - 3.461 x 106 mm/sec. This problem is similar to that"

addressed by Broberg [131, except that Broberg treated the crack as opening

from a zero initial length. The mesh used for the problem is shown in Figure

9. The model employed 213 nodes and 60 eight-noded isoparametric elements.

The time integration employed a time step of 0.2887 microsecond with a - 0.25

and 6 - 0.5 The results of the analysis is shown in Figure 10 Plong with the

Broberg solution and solutions of Atluri [14]. The figure shows that the two

computed solutions are very similar and they converge to the Broberg solution

after the initial transient conditions have past. Hence, very reasonable

estimates of the stress intensity factor with time can be obtained for the

problem of a crack with prescribed velocity.

The inverse of the problem performed above is perhaps of Gven more

importance; i.e., a problem in which the dynamic fracture toughness is speci-

fied and the crack length time history is sought. The ability of the proposed

procedure to perform such an analysis was tested by comparing computed results

with experimental data for a quasi-statically initiated crack in a 4340 steel

three-point bend (dynamic tear) specimen; see Figure 11. The finite element

mesh used for the analysis used 314 nodes and 91 eight-noded isoparametric

elements; see Figure 12. A previously determined dynamic fracture toughness

relation given by KID - 65 + 0.44 V was used [15].

The results of the analysis given in Figure 13 show excellent agree-

ment with experimental results.

-- ____ - A-4I

15 1II

---- - - - -- - -1

-4

K I

ii1 4

-I - - - - - - - 4

.11�-�--�� - -r Initial Crock Tip

K FIGURE 9 * �SH FOR A CONSTANT VELOCITY CRACK PRO3LEM

AL

16

1.5 BrobergProse /W'20.2

.01

II

1.0[ --I -.o , }

0.- ri5I,=

oI I ,.0d.2 0.3 0.4 0.5

0/W

FIGURE 10. STRESS INTENSITY FACTOR FOR A CRACK STARTING AT a /W - 0.2AND PROPAGATING AT A CONSTANT VELOCITY OF V/CS 8 .2.

17

iII

Support Support I

, i

LL

W-0.Loadl Point i

FIGURE 11. SPECIMEN GEOMETRY FOR A QUASI-STATICALLY INITIATEDCRACK PROPAGATION TEST L - 181 m, W 38 mm,B = 15.8 -m, W-a - 28.5 m.

18 *

I

_.•' O~r••n ~ bc~o o • OAD OIN

- .''/,,I, '' ' ,,, '

Initit Crck Poitio

FIUR 2.MS-FO MPC LODE TE-PNTBDANLYIS

1919

30 -

/ � sComputotion Stopped

i/20-A

i.A

10t A Experiment- Elastodynomic F.E. Solution

KID-65", .044 V (MPo.,n)

00 50 100

Time ý. seconds

FIGURE 13. A COMPARISON OF ANALYSIS AND EXPERIMEVT FOR AQUASI-STATICALLY INITIATED CRACK IN A 4340 DYNAMICTEAR SPECIMEN.

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CONCLUS IONS

A simple yet effective technique has been developed to performdynamic crack propagation problems. The technique does not require using

singular sleu, nta or updating the finite element mesh during crack propaga-

tion. Also, the dynamic fracture criterion need not be velocity dependent.

Good agreement has been obtained between analytical results and experimentaldata.

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ACKNOWLEDGEMENT i

The research described-in this paper was supported by the

Structural Mechanics Program of the Office of Naval Research under Contract

Number N00014-77-C-0576. The authors would like to express theirappreciation to Drs. Nicholas Perrone and Yapa Rajapakse of ONR for theirencouragement of this work.

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REFERENCES

[11 Kanninen, M. F., "Whither Dynamic Fracture Mechanics?", Numerical iMethods in Fracture Mechanics, Proceedings of the Second InterrationalConference huid at University College, Swansea, United Kingdom, Ede.D.R.J. Owen, A. R. Luxmoore, July 1980.

[2] Klaus-Jurgen, Bathe, and Wilson, E. L., Numerical Methods In FiniteElement Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey,pp T22-3126, 1976.

[3] Aoki, S., Kishimoto, K., Kondo, H., and Sakata, M., "ElastodynamicAnalysis of Cracks by Finite Element Method Using Singular Element",Int. J. Fracture, 14, pp 59-68, 1978.

(4] Bazant, Z. 0., Glauik, J. L., Jr., and Achenbach, J. D., "Finite ElementAnalysis of Wave Diffraction by 4 Crack", Mech. Div. ASCE, 102-EM3,pp 479-496, 1976.

(5] Aberson, J. A., Anderson, J. M., and King, W. W., _Dna alysis ofCracked Structures Using Singularity Finite Elements, 4astodynamc

Crack Problems, pp 249-294, Ed. G. C. Sih, Noordhoff, 1977.

[6] Kishimoto, K., Aoki, S., and Sakata, M., "Dynamic Stress Intensity

Factors Using J-Integral and Finite Element Method", EngineeringFracture Mechanics, 13, pp 387-394.

(7] Mall, S., "Finite Element Analysis of Stationary Cracks in TimeDependent Stress Fields", Numerical Methods in Fracture Mechanics, Eds.A. R. Luxmoore and D.R.J. Owen, University College, Swansea, 1980.

[8] Chen, Y. M., "Numerical Computation of Dynamic Stress Intensity Factorsby a Lagrangian Finite-Difference Method (The Hemp Code)", EngineeringFracture Mechanics, 7, pp 653-660, 1975.

[9] Popelar, C. H., and Gehlen, P. C., "Modeling of Dynamic CrackPropagation: II. Validation of Dynamic Analysis", InternationalJournal of Fracture, 15, pp 159-177, 1979.

[10] Mall, S., and Luz, J., "Use of an Eight-Node Element for Fast FractureProblems", International Journal of Fracture, 16, pp R33-R36, 1980.

[11] Mulluck, J. F., and King, W. W., "Fast Fracture Simulated byConventional Finite Elements: A Comparison of Two Energy-ReleaseAlgorithms", Crack Arrest Methodology and Applications, ASTM STP 711,Eds. G. T. Hahn and M. F. Kanninen, American Society for Testing andMaterials, pp 138-153, 1980.

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[121 Kobayashi, A. S., Urabe, Y., Mark, S., Emery, A. F., and Love, W. J.,"Dynamic Finite Element Analysis of Two Compact Specimens", Journal ofEngineering Materials and Technology, 1.00, pp 402-410, Octobtr 1978.

[13] Broberg, K. B., "The Propagation of a Brittle Crack", Arkiv for Fysik,Band 18, No. 10, pp 139-192, 1966.

[14] Atluri, S. N., Niahioka, T., and Nakagaki, M., "Numerical Modeling ofDynamic and Nonlinear Crack Propagation in Finite Bodies by MovingSingular Elements", Nonlinear and Dynamic Fracture Mechanics, Eds.Nicholas Perrone and Satya Atluri, American Society of MechanicalEngineers, AMD-Vol 35, pp 37-66, 1979.

[151 Kanninen, M. F., Gehlen, P. C., Barnes, C. R., Hoagland, R. G., Hahn,G. T., and Popelar, C. H., "Dynamic Crack Propagation Under ImpactLoading", Nonlinear and Dynamic Fracture Mechanics, Edo. NicholasPerrone and Satya Atluri, American Society of Mechanical Engineers, AMD-Vol 35, pp 195-200, 1979.

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